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I have data to fit a one-way ANOVA model to categorical data. I used a simple model first, getting significant coefficients, but with an $R^2$ of 0.2.

I then tried a log model and got more significant coefficients, but the $R^2$ is still only 0.3.

Is this normal for categorical data? Are there any other methods I should try to improve the model?

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R squared is not an absolute measure to trust, there is no real indication stating whether it is low or high enough. Then, you should pay attention to how you interpret differences in R squared among models: if your response variable in your second model is transformed (it's a log), you cannot compare R squared with the previous model.

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